- Volume 52 Issue 2
By the works of Kondo and Sakai, it is known that Alexander polynomials of knots which are transformed into the trivial knot by a single crossing change are characterized. In this note, we will characterize Alexander polynomials of knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change.
Alexander polynomials;Crossing change;Trefoil knot;Figure-eight knot
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts Math., 84, Second Edition, Springer-Verlag, New York, 1990.
- H. Kondo, Knots of unknotting number 1 and their Alexander polynomials, Osaka J. Math., 16(1979), 551-559
- J. Levine, A characterization of knot polynomials, Topology, 4(1965), 135-141. https://doi.org/10.1016/0040-9383(65)90061-3
- Y. Nakanishi, Local moves and Gordian complexes, II, Kyungpook Math. J., 47(2007), 329-334.
- D. Rolfsen, A surgical view of Alexander's polynomial, in Geometric Topology (Proc. Park City, 1974), Lecture Notes in Math. 438, Springer-Verlag, Berlin and New York, 1974, pp. 415-423.
- D. Rolfsen, Knots and Links, Math. Lecture Series 7, Publish or Perish Inc., Berkeley, 1976.
- T. Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ., 5(1977), 451-456.
- H. Seifert, Uber das Geschlecht von Knoten, Math. Ann., 110(1934), 571-592.
- T. Takagi, Shotou Seisuuron Kougi (in Japanese) [Lectures on Elementary Number Theory], Second Edition, Kyoritsu Shuppan, Tokyo, 1971.